The conjugate partition is obtained by reflecting the Ferrers graph
across the main diagonal or, equivalently, by representing each integer by a
column of dots. The conjugate to the example in Figure 26.9.1
is 6+5+4+2+1+1+1. Conjugation establishes a one-to-one correspondence between
partitions of n into at most k parts and partitions of n into parts with
largest part less than or equal to k. It follows that pk⁡(n)
also equals the number of partitions of n into parts that are less than
or equal to k.

pk⁡(≤m,n) is the number of partitions of n into at most k parts, each
less than or equal to m. It is also equal to the number of lattice
paths from (0,0) to (m,k) that have exactly n vertices (h,j),
1≤h≤m, 1≤j≤k, above and to the left of the lattice
path. See Figure 26.9.2.